The Sun's gravitational pull on the Moon is always stronger than the Earth's. (Do the math and see--on average the Sun's pull is more than twice as strong as the Earth's.) So the net acceleration of the Moon is always towards the Sun, never towards the Earth.
Here’s a visualisation https://www.youtube.com/watch?v=KBcxuM-qXec
The Sun's gravitational pull on the Moon is always stronger than the Earth's. (Do the math and see--on average the Sun's pull is more than twice as strong as the Earth's.) So the net acceleration of the Moon is always towards the Sun, never towards the Earth.